Abstract

Just as a rotating magnetised neutron star has material pulled away
from its surface to populate a magnetosphere, a similar process can
occur as a result of neutron-star pulsations rather than
rotation. This is of interest in connection with the overall study
of neutron star oscillation modes but with a particular focus on the
situation for magnetars. Following a previous Newtonian analysis of
the production of a force-free magnetosphere in this way
Timokhin et al. (2000), we present here a corresponding
general-relativistic analysis. We give a derivation of the general
relativistic Maxwell equations for small-amplitude arbitrary
oscillations of a non-rotating neutron star with a generic magnetic
field and show that these can be solved analytically under the
assumption of low current density in the magnetosphere. We apply our
formalism to toroidal oscillations of a neutron star with a dipole
magnetic field and find that the low current density approximation
is valid for at least half of the oscillation modes, similarly to
the Newtonian case. Using an improved formula for the determination
of the last closed field line, we calculate the energy losses
resulting from toroidal stellar oscillations for all of the modes
for which the size of the polar cap is small. We find that general
relativistic effects lead to shrinking of the size of the polar cap
and an increase in the energy density of the outflowing
plasma. These effects act in opposite directions but the net result
is that the energy loss from the neutron star is significantly
smaller than suggested by the Newtonian treatment.

keywords:

Study of the internal structure of neutron stars (NSs) is of
fundamental importance for subatomic physics since these objects
provide a laboratory for studying the properties of high-density
matter under very extreme conditions. In particular, there is the
intriguing possibility of using NS oscillation modes as a probe for
constraining models of the equation of state of matter at supranuclear
densities. It was suggested long ago that if a NS is oscillating,
then traces of this might be revealed in the radiation which it emits
(Pacini & Ruderman, 1974; Tsygan, 1975; Boriakoff, 1976; Bisnovatyi-Kogan, 1995; Ding & Cheng, 1997; Duncan, 1998). Recently, a lot of interest has been focussed
on oscillations of magnetized NSs because of the discovery of
gamma-ray flare activity in Soft Gamma-Ray Repeaters (SGRs) which are
thought to be the very highly magnetised NSs known as magnetars
(for recent review on the SGRs see Woods & Thompson, 2006; Watts & Strohmayer, 2007).
The giant flares in these objects are thought to be powered by global
reconfigurations of the magnetic field and it has been suggested that
the giant flares might trigger starquakes and excite global seismic
pulsations of the magnetar crust (Thompson & Duncan, 1995, 2001; Schwartz et al., 2005; Duncan, 1998). Indeed, analyses of the observations of
giant flares have revealed that the decaying part of the spectrum
exhibits a number of quasi-periodic oscillations (QPOs) with
frequencies in the range from a few tens of Hz up to a few hundred Hz
(Israel et al., 2005; Strohmayer & Watts, 2006; Watts & Strohmayer, 2006) and there has been a
considerable amount of theoretical effort attempting to identify these
with crustal oscillation modes (Glampedakis et al., 2006; Samuelsson & Andersson, 2007; Levin, 2007; Sotani et al., 2007a, b). While
there is substantial evidence that the observed SGR QPOs are caused by
neutron star pulsations, there is a great deal of uncertainty about
how stellar surface motion gets translated into the observed features
of the X-ray radiation (Strohmayer, 2008; Strohmayer & Watts, 2006; Timokhin et al., 2007). To make progress with this, it is necessary to
develop a better understanding of the processes occurring in the
magnetospheres of oscillating neutron stars.

Standard pulsars typically have magnetic fields of around 1012
G while magnetars may have fields of up to 1014−1015 G near to
the surface. Rotation of a magnetized star generates an electric
field:

Erot∼ΩRcB,

(1)

where B is the magnetic field strength, c is the speed of
light and Ω is the angular velocity of the star with radius R. Depending on the rotation velocity and the magnetic field
strength, the electric field may be as strong as 1010 V
cm−1 and it has a longitudinal component (parallel to B) which
can be able to pull charged particles away from the stellar surface,
if the work function is sufficiently small, and accelerate them up to
ultra-relativistic velocities. This result led Goldreich & Julian (1969) to
suggest that a rotating NS with a sufficiently strong magnetic field
should be surrounded by a magnetosphere filled with charge-separated
plasma which screens the accelerating electric field and thus hinders
further outflow of charged particles from the stellar surface. Even if
the binding energy of the charged particles is sufficiently high to
prevent them being pulled out by the electric field, the NS should
nevertheless be surrounded by charged particles produced by plasma
generation processes (Sturrock, 1971; Ruderman & Sutherland, 1975), which again
screen the longitudinal component of the electric field. These
considerations led to the development of a model for pulsar
magnetospheres which is frequently called the “standard model”
(an in depth discussion and review of this can be found in,
e.g., Michel, 1991; Beskin et al., 1993; Beskin, 2005).

Timokhin, Bisnovatyi-Kogan & Spruit (2000) (referred to as TBS from
here on) showed that an oscillating magnetized NS should also have a
magnetosphere filled with charge-separated plasma, even if it is not
rotating, since the vacuum electric field induced by the oscillations
would have a large radial component which can be of the same order as
rotationally-induced electric fields. One can show this quantitatively
by means of the following simple arguments. To order of magnitude, the
radial component of the vacuum electric field generated by the stellar
oscillations is given by

Eosc∼ωξcB,

(2)

where ω is the oscillation frequency and ξ is the
displacement amplitude. Using this together with Eq.(1), it
follows immediately that the electric field produced by oscillations
will be stronger than the rotationally induced one for sufficiently
slowly-rotating neutron stars, having

Ω≲ωξR.

(3)

For stellar oscillations with ξ/R∼0.001 and ω∼1 kHz,
the threshold is Ω∼1 Hz. Within this context, TBS developed a
formalism extending the basic aspects of the standard pulsar model to the
situation for a non-rotating magnetized NS undergoing arbitrary oscillations.
This formalism was based on the assumption of low current densities in the
magnetosphere, signifying that the influence of currents outside the NS on
electromagnetic processes occurring in the magnetosphere is negligibly small
compared to that of currents in the stellar interior. This assumption leads to
a great simplification of the Maxwell equations, which then can be solved
analytically. As an application of the formalism, TBS considered toroidal
oscillations of a NS with a dipole magnetic field, and obtained analytic
expressions for the electromagnetic field and charge density in the
magnetosphere. (Toroidal oscillations are thought to be particularly relevant
for magnetar QPO phenomena.) They found that the low current density
approximation (LCDA) is valid for at least half of all toroidal oscillation
modes and analyzed the energy losses due to plasma outflow caused by these
modes for cases where the size of the polar cap (the region on the stellar
surface that is crossed by open magnetic field lines) is small, finding that
the energy losses are strongly affected by the magnetospheric plasma. For
oscillation amplitudes larger than a certain critical value, they found that
energy losses due to plasma outflow were larger than those due to the emission
of the electromagnetic waves (assuming in that case that the star was
surrounded by vacuum). Recently, Timokhin (2007) considered spheroidal
oscillations of a NS with a dipole magnetic field, using the TBS formalism,
and found that the LCDA again holds for at least half of these modes.
Discussion in Timokhin (2007) also provided some useful insights into the
role of rotation for the magnetospheric structure of oscillating NSs.

The TBS model was a very important contribution and, to the best of
our knowledge, remains the only model for the magnetosphere of
oscillating NSs available in the literature. However, it should be
pointed out that it does not include several ingredients that a fully
consistent and realistic model ought to include. Most importantly, it
does not treat the magnetospheric currents in a fully consistent way:
although it gives a consistent solution for around half of the
oscillation modes, the remaining solutions turn out to be unphysical
and, as TBS pointed out, this is a symptom of the LCDA failing there.
Also, rotation and the effects of general relativity can be very
relevant; in particular, several authors have stressed that using a
Newtonian approach may not give very good results for the structure of
NS magnetospheres (see, e.g., Beskin, 1990; Muslimov & Tsygan, 1992; Mofiz & Ahmedov, 2000; Morozova et al., 2008). However, a more realistic model would naturally be more
complicated than the TBS one whose relative simplicity can be seen as
a positive advantage when using it as the basis for further
applications.

The aim of the present paper is to give a general relativistic
reworking of the TBS model so as to investigate the effects of the
changes with respect to the Newtonian treatment. We derive the general
relativistic Maxwell equations for arbitrary small-amplitude
oscillations of a non-rotating spherical NS with a generic magnetic
field configuration and show that they can be solved analytically
within the LCDA as in Newtonian theory. We then apply this solution to
the case of toroidal oscillations of a NS with a dipole magnetic
field and find that the LCDA is again valid for at least half of all
toroidal oscillation modes, as in Newtonian theory. Using an improved
formula for the determination of the last closed field line, we
calculate the energy losses resulting from these oscillations for
all of the modes for which the size of the polar cap is small
and discuss the influence of GR effects on the energy losses.

The paper is organized as follows. In Section 2 we introduce some definitions
and derive the quasi-stationary Maxwell equations in Schwarzschild spacetime as
well as the boundary conditions for the electromagnetic fields at the stellar
surface. In Section 3 we sketch our method for analytically solving the Maxwell
equations for arbitrary NS oscillations with a generic magnetic field
configuration. In Section 4 we apply our formalism to the case of purely
toroidal oscillations of a NS with a dipole magnetic field and also discuss the
validity of the LCDA and the role of GR effects. In Section 5 we calculate the
energy losses due to plasma outflow caused by the toroidal oscillations. Some
detailed technical calculations related to the discussion in the main part of
the paper are presented in Appendices A-C.

We use units for which c=1, a space-like signature (−,+,+,+) and
a spherical coordinate system (t,r,θ,ϕ). Greek indices are
taken to run from 0 to 3 while Latin indices run from 1 to 3 and we
adopt the standard convention for summation over repeated indices. We
indicate four-vectors with bold symbols (e.g.u)
and three-vectors with an arrow (e.g.→u).

2.1 Quasi-stationary Maxwell equations in Schwarzschild
spacetime

The study of electromagnetic processes related to stellar oscillations
in the vicinity of NSs should, in principle, use the coupled system of
Einstein-Maxwell equations. However, such an approach would be overly
complicated for our study here, as it is for many other astrophysical
problems. Here we simplify the problem by neglecting the contributions
of the electromagnetic fields, the NS rotation and the NS oscillations
to the spacetime metric and the structure of the NS3, noting
that this is expected to be a good approximation for small-amplitude
oscillations. Indeed, for a star with average mass-energy density ¯ρ, mass M and radius R, the maximum fractional change in
the spacetime metric produced by the magnetic field is typically of
the same order as the ratio between the energy density in the surface
magnetic field and average mass-energy density of the NS, i.e.,

B28π¯ρc2≃10−7(B1015G)2(1.4M⊙M)(R10km)3.

(4)

The corresponding fractional change in the metric due to rotation is
of order

0.1(ΩΩK)2=10−7(Ω1Hz)2(1kHzΩK)2

(5)

where ΩK is the Keplerian angular velocity at the
surface of the NS. Moreover, in the case of magnetars, which we
consider in our study, the oscillations are thought to be triggered by
the global reconfiguration of the magnetic field. Due to this reason, the
corrections due to the oscillations should not exceed the contribution
due to the magnetic field itself given by estimate
(4). Therefore, we can safely work in the
background spacetime of a static spherical star, whose line element in
a spherical coordinate system (t,r,θ,ϕ) is given by

ds2=g00(r)dt2+g11(r)dr2+r2dθ2+r2sin2θdϕ2,

(6)

while the geometry of the spacetime external to the star (i.e. for r≥R) is given by the Schwarzschild solution:

ds2=−N2dt2+N−2dr2+r2dθ2+r2sin2θdϕ2,

(7)

where N≡(1−2M/r)1/2 and M is the total mass of the
star. For the part of the spacetime inside the star, we represent the
metric in terms of functions Λ and Φ as

g00=−e2Φ(r),g11=e2Λ(r)=(1−2m(r)r)−1,

(8)

where m(r)=4π∫r0r′2ρ(r′)dr′ is the volume integral of the
total energy density ρ(r) over the spatial coordinates. The form of these
functions is given by solution of the standard TOV equations for spherical
relativistic stars (see, e.g., Shapiro & Teukolsky, 1983) and they are matched
continuously to the external Schwarzschild spacetime through the relations

g00(r=R)=N2R,g11(r=R)=N−2R,

(9)

where NR≡(1−2M/R)1/2. Within the external
part of the spacetime, we select a family of static observers with
four-velocity components given by

(uα)obs≡N−1(1,0,0,0).

(10)

and associated orthonormal frames having tetrad four
vectors {e^μ}=(e^0,e^r,e^θ,e^ϕ) and 1-forms {ω^μ}=(ω^0,ω^r,ω^θ,ω^ϕ), which will become useful when
determining the “physical” components of the electromagnetic
fields. The components of the vectors are given by equations (6)-(9)
of Rezzolla & Ahmedov (2004) (hereafter Paper I).

The general relativistic Maxwell equations have the following form
(Landau & Lifshitz, 1987)

3F[αβ,γ]=Fαβ,γ+Fγα,β+Fβγ,α=0,

(11)

Fαβ;β=4πJα,

(12)

where Fαβ is the electromagnetic field tensor and
J is the electric-charge 4-current. We consider the
region close to the star (the near zone), at distances from the NS
much smaller than the wavelength λ=2πc/ω. In
the near zone the electromagnetic fields are quasi-stationary,
therefore we neglect the displacement current term in the Maxwell
equations. Once expressed in terms of the physical components of the
electric and magnetic fields, equations (11) and
(12) become (see Section 2 of Paper I for
details of the derivation)

sinθ∂r(r2B^r)+N−1r∂θ(sinθB^θ)+N−1r∂ϕB^ϕ=0,

(13)

(rsinθ)∂B^r∂t

=

N[∂ϕE^θ−∂θ(sinθE^ϕ)],

(14)

(N−1rsinθ)∂B^θ∂t

=

−∂ϕE^r+sinθ∂r(rNE^ϕ),

(15)

(N−1r)∂B^ϕ∂t

=

−∂r(rNE^θ)+∂θE^r,

(16)

Nsinθ∂r(r2E^r)+r∂θ(sinθE^θ)+r∂ϕE^ϕ=4πρer2sinθ,

(17)

[∂θ(sinθB^ϕ)−∂ϕB^θ]

=

4πrsinθJ^r,

(18)

∂ϕB^r−sinθ∂r(rNB^ϕ)

=

4πrsinθJ^θ,

(19)

∂r(NrB^θ)−∂θB^r

=

4πrJ^ϕ,

(20)

where ρe is the proper charge density. We further assume that
the force-free condition,

→ESC⋅→B=0,

(21)

is fulfilled everywhere in the magnetosphere, implying that the
magnetosphere of the NS is populated with charged particles that
cancel the longitudinal component of the electric field. The charge
density ρSC responsible for the electric field →ESC (cf. equation 17) is the characteristic
charge density of the force-free magnetosphere; this is appropriate
for describing the charge density in the inner parts of the NS
magnetosphere. We will refer to →ESC as the
space-charge (SC) electric field, while to ρSC as the
SC charge density.

Finally, we introduce the perturbation of the NS crust in terms of its
four-velocity, with the components being given by

Missing or unrecognized delimiter for \bigg

(22)

where δvi=dxi/dt is the relative oscillation three-velocity
of the conducting stellar surface with respect to the unperturbed
state of the star.

2.2 Boundary conditions at the surface of star

We now begin our study of the internal electromagnetic field induced
by the stellar oscillations. We assume here that the material in the
crust can be treated as a perfect conductor and the induced electric
field then depends on the magnetic field and the pulsational velocity
field according to the following relations (see Paper I for details of
the derivation):

E^rin=−e−Φ[δv^θB^ϕ−δv^ϕB^θ],

(23)

E^θin=−e−Φ[δv^ϕB^r−δv^rB^ϕ],

(24)

E^ϕin=−e−Φ[δv^rB^θ−δv^θB^r].

(25)

Boundary conditions for the magnetic field at the stellar surface
(r=R) can be obtained from the requirement of continuity for the
radial component, while leaving the tangential components free to be
discontinuous because of surface currents:

B^rex|r=R=B^rin|r=R,

(26)

B^θex|r=R=B^θin|r=R+4πi^ϕ,

(27)

B^ϕex|r=R=B^ϕin|r=R−4πi^θ,

(28)

where i^i is the surface current density. Boundary
conditions for the electric field at the stellar surface are obtained
from requirement of continuity of the tangential components, leaving
E^r to have a discontinuity proportional to the surface
charge density Σs:

E^rex|r=R

=

E^rin|r=R+4πΣs=−N−1R[δv^θB^ϕ−δv^ϕB^θ]|r=R+4πΣs,

(29)

E^θex|r=R

=

E^θin|r=R=−N−1R[δv^ϕB^r−δv^rB^ϕ]|r=R,

(30)

E^ϕex|r=R

=

E^ϕin|r=R=−N−1R[δv^rB^θ−δv^θB^r]|r=R,

(31)

where Σs is the surface charge density.

2.3 The low current density approximation

The low current density approximation was introduced by TBS, and in
the present section we present a brief introduction to it for
completeness. Close to the NS surface, the current flows along the
magnetic field lines, and so in the inner parts of the magnetosphere
it can be expressed as

→J=α(r,θ,ϕ)⋅→B,

(32)

where α is a scalar function. The system of equations
(13)–(20), (21) and (32)
forms a complete set but is overly complicated for solving in the
general case. However, within the LCDA these equations can, as we show
below, be solved analytically for arbitrary oscillations of a NS with
a generic magnetic field configuration.

The LCDA scheme is based on the assumption that the perturbation of
the magnetic field induced by currents flowing in the NS interior is
much larger than that due to currents in the magnetosphere, which are
neglected to first order in the oscillation parameter
¯ξ≡ξ/R:

4πc→J≪∇×→B,

(33)

and

∇×→B(1)=0,

(34)

where →B(1) is the first order term of the expansion in ¯ξ. This also implies that the current density satisfies the
condition

J≪1r(B(0)ξR)c≈ρSC(R)c(cωr),

(35)

where ρSC(R) is the SC density near to the surface of
the star. Here we have used the relation ρSC(R)≃B(0)η/cR, where η is the velocity amplitude of the
oscillation and ω is its frequency.

In regions of complete charge separation, the maximum current density
is given by ρSCc. Since the absolute value of
ρSC decreases with increasing r and because r≪c/w in the near zone, condition (35) is satisfied in the
magnetosphere if there is complete charge separation there. Since the
current in the magnetosphere flows along magnetic field lines, its
magnitude does not change and so condition (35) is also
satisfied along magnetic field lines in non-charge-separated regions
as long as they have crossed regions with complete charge separation.

In the following, we solve the Maxwell equations assuming that
condition (35) is satisfied throughout the whole near zone. As
discussed above, a regular solution of the system of equations
(13)-(20), (21) and (32)
should exist for arbitrary oscillations and arbitrary configurations
of the NS magnetic field and so, as shown by TBS, if a solution has an
unphysical behaviour, this would imply that the LCDA fails for this
oscillation and that the accelerating electric field cannot be
screened only by a stationary configuration of the charged-separated
plasma. In some regions of the magnetosphere, the current density
could be as high as

J≃ρSCc(cωr).

(36)

For a more detailed discussion of the LCDA and its validity, we refer
the reader to Sections 2.3 and 3.2.1 of TBS.

3.1 The electromagnetic field in the magnetosphere

We now begin our solution of the Maxwell equations, assuming that the
LCDA condition (35) is satisfied everywhere in the
magnetosphere. Within the LCDA, equations (18)-(20)
for the magnetic field in the magnetosphere take the form

∂θ(sinθB^ϕ)−∂ϕB^θ

=

0,

(37)

∂ϕB^r−sinθ∂r(rNB^ϕ)

=

0,

(38)

∂r(NrB^θ)−∂θB^r

=

0.

(39)

As demonstrated in Paper I, the components of the magnetic field
B^r,B^θ and B^ϕ can be expressed in
terms of a scalar function S in the following way:

B^r

=

−1r2sin2θ[sinθ∂θ(sinθ∂θS)+∂ϕ∂ϕS],

(40)

B^θ

=

Nr∂θ∂rS,

(41)

B^ϕ

=

Nrsinθ∂ϕ∂rS.

(42)

Substituting these expressions into the Maxwell equations
(14)–(16), we obtain a system of equations for the
electric field components which has the following general solution

E^rSC=−∂r(ΨSC),

(43)

E^θSC=−1Nrsinθ∂t∂ϕS−1Nr∂θ(ΨSC),

(44)

E^ϕSC=1Nr∂t∂θS−1Nrsinθ∂ϕ(ΨSC),

(45)

where ΨSC is an arbitrary scalar function. The terms
proportional to the gradient of ΨSC are responsible for
the contribution of the charged particles in the magnetosphere. The
vacuum part of the electric field is given by the derivatives of the
scalar function S. Substituting (43)-(45) into
equation (17), we get an expression for the SC charge density
in terms of ΨSC:

ρSC=−14πr2[N∂r(r2∂rΨSC)+1N△ΩΨSC],

(46)

where △Ω is the angular part of the Laplacian:

△Ω=1sinθ∂θ(sinθ∂θ)+1sin2θ∂ϕϕ.

(47)

3.2 The equation for ΨSC

Substituting expressions (40)–(42) and
(43)–(45) for the components of the electric and
magnetic fields into the force-free condition (21), we
get the following equation for ΨSC

1sin2θ[sinθ∂θ(sinθ∂θS)+∂ϕ∂ϕS]∂r(ΨSC)

−

1sinθ[∂ϕ∂tS∂θ∂rS−∂θ∂tS∂ϕ∂rS]

(48)

−

∂θ∂rS∂θ(ΨSC)−1sin2θ∂ϕ∂rS∂ϕ(ΨSC)=0.

If the amplitude of the NS oscillations is suitably small (¯ξ≪1), the function S can be series expanded in terms of the
dimensionless perturbation parameter ¯ξ and can be
approximated by the sum of the two lowest order terms

S(t,r,θ,ϕ)=S0(r,θ,ϕ)+δS(t,r,θ,ϕ).

(49)

Here the first term S0 corresponds to the unperturbed static
magnetic field of the NS, while δS is the first order
correction to it. At this level of approximation, equation (48)
for ΨSC takes the form

1sin2θ[sinθ∂θ(sinθ∂θS0)+∂ϕ∂ϕS0]∂r(ΨSC)

−

1sinθ[∂ϕ∂t(δS)∂θ∂rS0−∂θ∂t(δS)∂ϕ∂rS0]

(50)

−

∂θ∂rS0∂θ(ΨSC)−1sin2θ∂ϕ∂rS0∂ϕ(ΨSC)=0.

Next we expand S in terms of the spherical harmonics:

S=∞∑ℓ=0ℓ∑m=−ℓSℓm(t,r)Yℓm(θ,ϕ).

(51)

where the functions Sℓm are given in terms of Legendre
functions of the second kind Qℓ by (Rezzolla et al., 2001)

Sℓm(t,r)=−r2M2ddr[r(1−2Mr)ddrQℓ(1−rM)]sℓm(t).

(52)

Note that all of the time dependence in (52) is contained in
the integration constants sℓm(t) which, as we will see later,
are determined by the boundary conditions at the surface of the star.
We now series expand the coefficients Sℓm(t,r) and sℓm(t) in terms of ¯ξ

Sℓm(t,r)=S0ℓm(r)+δSℓm(t,r),sℓm(t)=s0ℓm+δsℓm(t),

(53)

where all of the time dependence is now confined within the
coefficients δSℓm(r,t) and δsℓm(t),
while the coefficients S0ℓm and s0ℓm are responsible
for the unperturbed static magnetic field of the star. Using these
results, we can also express S and δS in terms of a series
in Yℓm(θ,ϕ) in the following way

S0

=

∞∑ℓ=0ℓ∑m=−ℓS0ℓm(r)Yℓm(θ,ϕ),

(54)

δS

=

∞∑ℓ=0ℓ∑m=−ℓδSℓm(t,r)Yℓm(θ,ϕ).

(55)

The variables r and t in the functions Sℓm(t,r) and
Sℓm(t,r) can be separated using relation (52):

S0ℓm(r)

=

Missing or unrecognized delimiter for \right

(56)

Sℓm(t,r)

=

Missing or unrecognized delimiter for \right

(57)

3.3 The boundary condition for ΨSC

We now derive a boundary condition for ΨSC at the
stellar surface using the behaviour of the electric and magnetic
fields in that region. Following TBS, we assume that near to the
stellar surface the interior magnetic field has the same behaviour as
the exterior one:

B^r=−C1r2sin2θ[sinθ∂θ(sinθ∂θS)+∂ϕϕS],

(58)

B^θ=C1e−Λr∂θ∂rS,

(59)

B^ϕ=C1e−Λrsinθ∂ϕ∂rS.

(60)

Using the continuity condition for the normal component of the
magnetic field [B^r]=0 at the stellar surface
(Pons & Geppert, 2007) together with the condition e−Λ|r=R≡NR, one finds that the integration constant C1 is equal to
one. The interior electric field components can then be obtained by
substituting (58) – (60) (with C1=1) into
(23) – (25):

E^rin

=

−e−(Φ+Λ)rsinθ{δv^θ∂ϕ∂rS−sinθδv^ϕ∂θ∂rS},

(61)

E^θin

=

e−(Φ+Λ)rsinθ{δv^r∂ϕ∂rS+δv^ϕeΛrsinθ[sinθ∂θ(sinθ∂θS)+∂ϕ∂ϕS]},

(62)

E^ϕin

=

−e−(Φ+Λ)r{δv^r∂θ∂rS+δv^θeΛrsin2θ[sinθ∂θ(sinθ∂θS)+∂ϕ∂ϕS]}.

(63)

The continuity condition for the θ component of the electric
field across the stellar surface (30) gives a boundary
condition for ∂θΨSC|r=R:

while the continuity condition for E^ϕ (31)
gives a boundary condition for ∂ϕΨSC|r=R:

Missing or unrecognized delimiter for \right

(65)

Integration of equation (64) or equation
(65) over θ or ϕ respectively, gives a
boundary condition for ΨSC. We will use the result of
integrating equation (64) over θ. Assuming that
the perturbation depends on time t as e−iωt, we obtain
the following condition, correct to first order in ¯ξ,

The components of the stellar-oscillation velocity field are
continuously differentiable functions of r,θ and ϕ.
The boundary conditions for the electric field
(30)-(31) imply that the tangential components
of the electric field →ESC must be finite. The
vacuum terms on the right-hand side of (44)-(45)
and the terms on both sides of equation (45) are also
finite. Consequently, the term

−∂ϕ(ΨSC)sinθ|r=R

(67)

should also be finite. Hence we obtain that ∂ϕ(ΨSC)|θ=0,π;r=R=0 and so the
function F(ϕ) in the expression for boundary condition
(66) must satisfy the condition (ΨSC)|θ=0,π;r=R=Ce−iωt, where C
is a constant. Using gauge invariance, we choose

ΨSC|θ=0;r=R=0,

(68)

and from this and equation (66), we obtain our
expression for the boundary condition for ΨSC at the
stellar surface:

As an important application of this formalism, we now consider
small-amplitude toroidal oscillations of a NS with a dipole magnetic
field. For toroidal oscillations in the (ℓ′,m′) mode, a
generic conducting fluid element is displaced from its initial
location (r,θ,ϕ) to a perturbed location
(r,θ+ξθ,ϕ+ξϕ) with the velocity field
(Unno et al., 1989),

where ω is the oscillation frequency and η(r) is the
transverse velocity amplitude. Note that in the above expressions
(70), the oscillation mode axis is directed along the
z-axis. We use a prime to denote the spherical harmonic indices in
the case of the oscillation modes.

4.1 The unperturbed exterior dipole magnetic field

If the static unperturbed magnetic field of the NS is of a dipole
type, then the coefficients s0ℓm involved in specifying it
have the following form (see eq. 117 of Paper I)

s010=−√3π2μcosχ,s011=√3π2μsinχ,

(71)

where μ is the magnetic dipole moment of the star, as measured
by a distant observer, and χ is the inclination angle between the
dipole moment and z-axis. Substituting expressions (71) into
(56) and then the latter into (54), we get

S0=−3μr28M3[lnN2+2Mr(1+Mr)](cosθcosχ+eiϕsinθsinχ)

(72)

The corresponding magnetic field components have the form

B^r0=−3μ4M3[lnN2+2Mr(1+Mr)](cosχcosθ+sinχsinθeiϕ),

(73)

B^θ0=3μN4M2r[rMlnN2+1N2+1](cosχsinθ−sinχcosθeiϕ),

(74)

B^ϕ0=3μN4M2r[rMlnN2+1N2+1](−isinχeiϕ).

(75)

At the stellar surface, these expressions for the unperturbed
magnetic field components become

where B0 is defined as B0=2μ/R3. In Newtonian theory
B0 would be the value of the magnetic strength at the magnetic
pole but this becomes modified in GR. The GR modifications are
contained within the parameters

hR=3R2NR8M2[RMlnN2R+1N2R+1],fR=−3R38M3[lnN2R+2MR(1+MR)].

(77)

For a given μ, the magnetic field near to the surface of the NS
is stronger in GR than in Newtonian theory, as already noted by
Ginzburg & Ozernoy (1964).

4.2 The equation for ΨSC

Substituting S0 from (72) into equation (50), we
obtain a partial differential equation containing two unknown
functions ΨSC and δS for arbitrary oscillations
of a NS with a dipole magnetic field

Using the expressions for the velocity field of the toroidal
oscillations (70) and for the boundary conditions for
the partial derivatives of the SC potential
(64)-(65), we find that
∂tδS is given by (see Appendix A for details of the
derivation)

∂tδS(r,t)

=

∞∑ℓ=0ℓ∑m=−ℓB0RfR~ηRℓ(ℓ+1)r2qℓ(r)R2qℓ(R)

(80)

×

∫4π[∂θYℓm(sinθcosχ−eiϕcosθsinχ)+ieiϕ∂ϕYℓmsinθsinχ]Y∗ℓ′m′(θ,ϕ)sinθdΩ.

From here on, for simplicity, we will consider only the case with χ=0. Although our solution depends on the angle between the
magnetic field axis and the oscillation mode axis, focusing on the
case χ=0 does not actually imply a loss of generality because
any mode with its axis not aligned with a given direction can be
represented as a sum of modes with axes along this direction. We have
developed a MATHEMATICA code for analytically solving equation
(50) and hence obtaining analytic expressions for the
electric and magnetic fields and for the SC density.

The solution of equation (78) for the case χ=0 is
given in Appendix B, where we show
that the general solution has the following form